Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > sge0ad2en | Structured version Visualization version GIF version |
Description: The value of the infinite geometric series 2↑-1 + 2↑-2 +... , multiplied by a constant. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
Ref | Expression |
---|---|
sge0ad2en.1 | ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
sge0ad2en | ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfv 1843 | . 2 ⊢ Ⅎ𝑛𝜑 | |
2 | 0xr 10086 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ∈ ℝ*) |
4 | pnfxr 10092 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → +∞ ∈ ℝ*) |
6 | rge0ssre 12280 | . . . . . . 7 ⊢ (0[,)+∞) ⊆ ℝ | |
7 | sge0ad2en.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ (0[,)+∞)) | |
8 | 6, 7 | sseldi 3601 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
9 | 8 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝐴 ∈ ℝ) |
10 | 2re 11090 | . . . . . . 7 ⊢ 2 ∈ ℝ | |
11 | 10 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ) |
12 | nnnn0 11299 | . . . . . . 7 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0) | |
13 | 12 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ0) |
14 | 11, 13 | reexpcld 13025 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ) |
15 | 2cnd 11093 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℂ) | |
16 | 2ne0 11113 | . . . . . . 7 ⊢ 2 ≠ 0 | |
17 | 16 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ≠ 0) |
18 | 13 | nn0zd 11480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ) |
19 | 15, 17, 18 | expne0d 13014 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ≠ 0) |
20 | 9, 14, 19 | redivcld 10853 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ) |
21 | 20 | rexrd 10089 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ ℝ*) |
22 | 2rp 11837 | . . . . . 6 ⊢ 2 ∈ ℝ+ | |
23 | 22 | a1i 11 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 2 ∈ ℝ+) |
24 | 23, 18 | rpexpcld 13032 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (2↑𝑛) ∈ ℝ+) |
25 | 2 | a1i 11 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ*) |
26 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → +∞ ∈ ℝ*) |
27 | icogelb 12225 | . . . . . 6 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,)+∞)) → 0 ≤ 𝐴) | |
28 | 25, 26, 7, 27 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝐴) |
29 | 28 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ 𝐴) |
30 | 9, 24, 29 | divge0d 11912 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → 0 ≤ (𝐴 / (2↑𝑛))) |
31 | 20 | ltpnfd 11955 | . . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) < +∞) |
32 | 3, 5, 21, 30, 31 | elicod 12224 | . 2 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴 / (2↑𝑛)) ∈ (0[,)+∞)) |
33 | 1zzd 11408 | . 2 ⊢ (𝜑 → 1 ∈ ℤ) | |
34 | nnuz 11723 | . 2 ⊢ ℕ = (ℤ≥‘1) | |
35 | 8 | recnd 10068 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
36 | eqid 2622 | . . . 4 ⊢ (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) = (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛))) | |
37 | 36 | geo2lim 14606 | . . 3 ⊢ (𝐴 ∈ ℂ → seq1( + , (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) ⇝ 𝐴) |
38 | 35, 37 | syl 17 | . 2 ⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) ⇝ 𝐴) |
39 | 1, 32, 33, 34, 38 | sge0isummpt 40647 | 1 ⊢ (𝜑 → (Σ^‘(𝑛 ∈ ℕ ↦ (𝐴 / (2↑𝑛)))) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 +∞cpnf 10071 ℝ*cxr 10073 ≤ cle 10075 / cdiv 10684 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ℝ+crp 11832 [,)cico 12177 seqcseq 12801 ↑cexp 12860 ⇝ cli 14215 Σ^csumge0 40579 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-sumge0 40580 |
This theorem is referenced by: ovnsubaddlem1 40784 ovolval5lem1 40866 |
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